二维时间分数阶扩散方程的Hermite型矩形元的超收敛分析
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摘要
基于经典的L1逼近,针对二维时间分数阶扩散方程给出Hermite型矩形元的全离散格式.首先,证明其逼近格式的无条件稳定性.其次,基于Hermite型矩形元的积分恒等式结果,建立插值与Ritz投影之间在H1模意义下的超收敛估计.进而,通过利用插值与投影的关系及巧妙地处理分数阶导数,得到单独利用插值或Ritz投影所无法得到的超逼近及超收敛结果.最后,借助于插值后处理技术导出了整体超收敛结果.
Based on the classical L1 approximation scheme, a Hermite-type rectangular element method is proposed for two-dimensional time fractional diffusion equations under the fully-discrete scheme.Firstly, unconditional stability analysis of the approcimate scheme is provided. Secondly, by use of the integral indentity result of Hermite-type rectangular element, superconvergence estimate in H1-norm is established between the interpolation and Ritz projection. Moreover, combining with the relationship between the interpolation operator and Ritz projection, and by dealing with fractional derivatives skillfully, superclose and superconvergence results are obtained, which cann't be deduced by interpolation or Ritz projection alone. Finally, the global superconvergence property is derived by the technique of the postprocessing operator.
Based on the classical L1 approximation scheme, a Hermite-type rectangular element method is proposed for two-dimensional time fractional diffusion equations under the fully-discrete scheme.Firstly, unconditional stability analysis of the approcimate scheme is provided. Secondly, by use of the integral indentity result of Hermite-type rectangular element, superconvergence estimate in H1-norm is established between the interpolation and Ritz projection. Moreover, combining with the relationship between the interpolation operator and Ritz projection, and by dealing with fractional derivatives skillfully, superclose and superconvergence results are obtained, which cann't be deduced by interpolation or Ritz projection alone. Finally, the global superconvergence property is derived by the technique of the postprocessing operator.
引文
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[32]石东洋,王芬玲,樊明智,等.Sine-Gordon方程的最低阶各向异性混合元高精度分析新途径[J].计算数学,2015,37(2):148-161.
[33]THOMEE V.Galerkin Finite Element Methods for Parabolic Problems[M].北京:世界图书出版公司北京公司,2003.
[2]YANG Q Q,TURNER I,LIU Fawang,et al.Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions[J].SIAM J.Sci.Comput.,2011,33(3):1159-1180.
[3]林世敏,许传炬.分数阶微分方程的理论和数值方法研究[J].计算数学,2016,38(01):1-24.
[4]ZHUANG P H,LIU F W,ANH V,et al.New solution and analytical techniques of the implicit numerical method for the anomalous sub-diffusion equation[J].SIAM J.Numer.Anal.,2008,46(2):1079-1095.
[5]CAO J Y,XU C J.A high order schema for the numerical solution of the fractional ordinary differential equations[J].J.Comput.Phys.,2013,238(2):154-168.
[6]LI C P,TAO C X.On the fractional Adams method[J].Comput.Math.Appl.,2009,58(8):1573-1588.
[7]JIANG Y J,MA J T.High-order finite element methods for time-fractional partial differential equations[J].J.Comput.Appl.Math.,2011,235(11):3285-3290.
[8]GAO G H,SUN Z Z,ZHANG Y N.A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions[J].J.Comput.Phys.,2012,231(7):2865-2879.
[9]ZHAO Y M,ZHANG Y D,LIU F W,et al.Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation[J].Appl.Math.Model.,2016,40(19-20):8810-8825.
[10]ZENG F H.Second-order stable finite difference schemes for the time-fractional diffusion-wave equation[J].J.Sci.Comput.,2015,65(1):411-430.
[11]MAHDY A,SWEILAM N,KHADER M.Crank-nicolson finite difference method for solving timefractional diffusion equation[J].J.Fract.Calc.Appl.,2012,2(2):1-9.
[12]ZHENG M L,LIU F W,ANH V,et al.A high-order spectral method for the multi-term time-fractional diffusion equations[J].Appl.Math.Model.,2016,40(7-8):4970-4985.
[13]DEHGHAN M,SAFARPOOR M,ABBASZADEH M.Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations[J].J.Comput.Appl.Math.,2015,290:174-195.
[14]JIN B T,LAZAROV R,LIU Y K,et al.The Galerkin finite element method for a multi-term timefractional diffusion equation[J].J.Comput.Phys.,2015,281:825-843.
[15]SHI Z G,ZHAO Y M,TANG Y F,et al.Superconvergence analysis of an H1-Galerkin mixed finite element method for two-dimensional multi-term time fractional diffusion equations[J].Int.J.Comput.Math.,2017:1-16.
[16]BU W P,LIU X T,TANG Y F,et al.Finite element multigrid method for multi-term time fractional advection diffusion equations[J].Int.J.Model.Sim.Sci.Comput.,2015,6(01):1540001.
[17]ZHAO Y M,ZHANG Y D,SHI D Y,et al.Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations[J].Appl.Math.Lett.,2016,59:38-47.
[18]LIU Y,DU Y W,LI H,et al.An H1-Galerkin mixed finite element method for time fractional reaction-diffusion equation[J].J.Appl.Math.Comput.,2015,47(1-2):103-117.
[19]ZHOU Z J,GONG W.Finite element approximation of optimal control problems governed by time fractional diffusion equation[J].Comput.Math.Appl.,2016,71(1):301-318.
[20]LIU Y,DU Y W,LI H,et al.A two-grid mixed finite element method for a nonlinear fourthorder reaction-diffusion problem with time-fractional derivative[J].Comput.Math.Appl.,2015,70(10):2474-2492.
[21]LIU Y,DU Y W,LI H,et al.Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem[J].Comput.Math.Appl.,2015,70(4):573-591.
[22]DEHGHAN M,ABBASZADEH M,MOHEBBI A.An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations[J].Eng.Anal.Bound.Elem.,2015,50(2):412-434.
[23]WEI L L,HE Y N,ZHANG X D,et al.Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schr¨odinger equation[J].Finite Elem.Anal.Des.,2012,59(4):28-34.
[24]石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推[J].计算数学,2005,27(4):369-382.
[25]王萍莉,史艳华,石东洋.广义神经传播方程的超收敛分析及外推[J].西北师范大学学报(自然科学版),2012,48(1):22-25.
[26]王萍莉,石东洋.一类非线性Schrodinger方程的非协调有限元分析[J].应用数学,2013,26(2):320-325.
[27]王萍莉,石东伟,王芬玲.四阶强阻尼波动方程新混合元模式的高精度分析[J].应用数学,2015,28(2):368-377.
[28]王萍莉,石东洋.广义神经传播方程的低阶H1-Galerkin混合元方法[J].数学的实践与认识,2015,45(10):229-237.
[29]王萍莉,石东洋.Schrodinger方程双线性元的超收敛分析和外推[J].山东大学学报(理学版),2014,49(10):66-71.
[30]SHI D Y,WANG P L,ZHAO Y M.Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrodinger equation[J].Appl.Math.Lett.,2014,38(38):129-134.
[31]石东洋,王芬玲,赵艳敏.非线性Sine-Gordon方程的各向异性线性元高精度分析新模式[J].计算数学,2014,36(3):245-256.
[32]石东洋,王芬玲,樊明智,等.Sine-Gordon方程的最低阶各向异性混合元高精度分析新途径[J].计算数学,2015,37(2):148-161.
[33]THOMEE V.Galerkin Finite Element Methods for Parabolic Problems[M].北京:世界图书出版公司北京公司,2003.
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